3.1669 \(\int \frac{(A+B x) (a^2+2 a b x+b^2 x^2)}{(d+e x)^5} \, dx\)

Optimal. Leaf size=86 \[ \frac{(a+b x)^3 (-4 a B e+A b e+3 b B d)}{12 e (d+e x)^3 (b d-a e)^2}-\frac{(a+b x)^3 (B d-A e)}{4 e (d+e x)^4 (b d-a e)} \]

[Out]

-((B*d - A*e)*(a + b*x)^3)/(4*e*(b*d - a*e)*(d + e*x)^4) + ((3*b*B*d + A*b*e - 4*a*B*e)*(a + b*x)^3)/(12*e*(b*
d - a*e)^2*(d + e*x)^3)

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Rubi [A]  time = 0.0345901, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {27, 78, 37} \[ \frac{(a+b x)^3 (-4 a B e+A b e+3 b B d)}{12 e (d+e x)^3 (b d-a e)^2}-\frac{(a+b x)^3 (B d-A e)}{4 e (d+e x)^4 (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2))/(d + e*x)^5,x]

[Out]

-((B*d - A*e)*(a + b*x)^3)/(4*e*(b*d - a*e)*(d + e*x)^4) + ((3*b*B*d + A*b*e - 4*a*B*e)*(a + b*x)^3)/(12*e*(b*
d - a*e)^2*(d + e*x)^3)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{(d+e x)^5} \, dx &=\int \frac{(a+b x)^2 (A+B x)}{(d+e x)^5} \, dx\\ &=-\frac{(B d-A e) (a+b x)^3}{4 e (b d-a e) (d+e x)^4}+\frac{(3 b B d+A b e-4 a B e) \int \frac{(a+b x)^2}{(d+e x)^4} \, dx}{4 e (b d-a e)}\\ &=-\frac{(B d-A e) (a+b x)^3}{4 e (b d-a e) (d+e x)^4}+\frac{(3 b B d+A b e-4 a B e) (a+b x)^3}{12 e (b d-a e)^2 (d+e x)^3}\\ \end{align*}

Mathematica [A]  time = 0.0626825, size = 125, normalized size = 1.45 \[ -\frac{a^2 e^2 (3 A e+B (d+4 e x))+2 a b e \left (A e (d+4 e x)+B \left (d^2+4 d e x+6 e^2 x^2\right )\right )+b^2 \left (A e \left (d^2+4 d e x+6 e^2 x^2\right )+3 B \left (4 d^2 e x+d^3+6 d e^2 x^2+4 e^3 x^3\right )\right )}{12 e^4 (d+e x)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2))/(d + e*x)^5,x]

[Out]

-(a^2*e^2*(3*A*e + B*(d + 4*e*x)) + 2*a*b*e*(A*e*(d + 4*e*x) + B*(d^2 + 4*d*e*x + 6*e^2*x^2)) + b^2*(A*e*(d^2
+ 4*d*e*x + 6*e^2*x^2) + 3*B*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3)))/(12*e^4*(d + e*x)^4)

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Maple [B]  time = 0.007, size = 166, normalized size = 1.9 \begin{align*} -{\frac{b \left ( Abe+2\,aBe-3\,Bbd \right ) }{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{A{a}^{2}{e}^{3}-2\,Adab{e}^{2}+A{d}^{2}{b}^{2}e-B{a}^{2}d{e}^{2}+2\,B{d}^{2}abe-{b}^{2}B{d}^{3}}{4\,{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{{b}^{2}B}{{e}^{4} \left ( ex+d \right ) }}-{\frac{2\,Aab{e}^{2}-2\,Ad{b}^{2}e+{a}^{2}B{e}^{2}-4\,Bdabe+3\,{b}^{2}B{d}^{2}}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)/(e*x+d)^5,x)

[Out]

-1/2*b*(A*b*e+2*B*a*e-3*B*b*d)/e^4/(e*x+d)^2-1/4*(A*a^2*e^3-2*A*a*b*d*e^2+A*b^2*d^2*e-B*a^2*d*e^2+2*B*a*b*d^2*
e-B*b^2*d^3)/e^4/(e*x+d)^4-b^2*B/e^4/(e*x+d)-1/3*(2*A*a*b*e^2-2*A*b^2*d*e+B*a^2*e^2-4*B*a*b*d*e+3*B*b^2*d^2)/e
^4/(e*x+d)^3

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Maxima [B]  time = 1.17545, size = 252, normalized size = 2.93 \begin{align*} -\frac{12 \, B b^{2} e^{3} x^{3} + 3 \, B b^{2} d^{3} + 3 \, A a^{2} e^{3} +{\left (2 \, B a b + A b^{2}\right )} d^{2} e +{\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 6 \,{\left (3 \, B b^{2} d e^{2} +{\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 4 \,{\left (3 \, B b^{2} d^{2} e +{\left (2 \, B a b + A b^{2}\right )} d e^{2} +{\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x}{12 \,{\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)/(e*x+d)^5,x, algorithm="maxima")

[Out]

-1/12*(12*B*b^2*e^3*x^3 + 3*B*b^2*d^3 + 3*A*a^2*e^3 + (2*B*a*b + A*b^2)*d^2*e + (B*a^2 + 2*A*a*b)*d*e^2 + 6*(3
*B*b^2*d*e^2 + (2*B*a*b + A*b^2)*e^3)*x^2 + 4*(3*B*b^2*d^2*e + (2*B*a*b + A*b^2)*d*e^2 + (B*a^2 + 2*A*a*b)*e^3
)*x)/(e^8*x^4 + 4*d*e^7*x^3 + 6*d^2*e^6*x^2 + 4*d^3*e^5*x + d^4*e^4)

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Fricas [B]  time = 1.44526, size = 392, normalized size = 4.56 \begin{align*} -\frac{12 \, B b^{2} e^{3} x^{3} + 3 \, B b^{2} d^{3} + 3 \, A a^{2} e^{3} +{\left (2 \, B a b + A b^{2}\right )} d^{2} e +{\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 6 \,{\left (3 \, B b^{2} d e^{2} +{\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 4 \,{\left (3 \, B b^{2} d^{2} e +{\left (2 \, B a b + A b^{2}\right )} d e^{2} +{\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x}{12 \,{\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)/(e*x+d)^5,x, algorithm="fricas")

[Out]

-1/12*(12*B*b^2*e^3*x^3 + 3*B*b^2*d^3 + 3*A*a^2*e^3 + (2*B*a*b + A*b^2)*d^2*e + (B*a^2 + 2*A*a*b)*d*e^2 + 6*(3
*B*b^2*d*e^2 + (2*B*a*b + A*b^2)*e^3)*x^2 + 4*(3*B*b^2*d^2*e + (2*B*a*b + A*b^2)*d*e^2 + (B*a^2 + 2*A*a*b)*e^3
)*x)/(e^8*x^4 + 4*d*e^7*x^3 + 6*d^2*e^6*x^2 + 4*d^3*e^5*x + d^4*e^4)

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Sympy [B]  time = 18.7575, size = 221, normalized size = 2.57 \begin{align*} - \frac{3 A a^{2} e^{3} + 2 A a b d e^{2} + A b^{2} d^{2} e + B a^{2} d e^{2} + 2 B a b d^{2} e + 3 B b^{2} d^{3} + 12 B b^{2} e^{3} x^{3} + x^{2} \left (6 A b^{2} e^{3} + 12 B a b e^{3} + 18 B b^{2} d e^{2}\right ) + x \left (8 A a b e^{3} + 4 A b^{2} d e^{2} + 4 B a^{2} e^{3} + 8 B a b d e^{2} + 12 B b^{2} d^{2} e\right )}{12 d^{4} e^{4} + 48 d^{3} e^{5} x + 72 d^{2} e^{6} x^{2} + 48 d e^{7} x^{3} + 12 e^{8} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)/(e*x+d)**5,x)

[Out]

-(3*A*a**2*e**3 + 2*A*a*b*d*e**2 + A*b**2*d**2*e + B*a**2*d*e**2 + 2*B*a*b*d**2*e + 3*B*b**2*d**3 + 12*B*b**2*
e**3*x**3 + x**2*(6*A*b**2*e**3 + 12*B*a*b*e**3 + 18*B*b**2*d*e**2) + x*(8*A*a*b*e**3 + 4*A*b**2*d*e**2 + 4*B*
a**2*e**3 + 8*B*a*b*d*e**2 + 12*B*b**2*d**2*e))/(12*d**4*e**4 + 48*d**3*e**5*x + 72*d**2*e**6*x**2 + 48*d*e**7
*x**3 + 12*e**8*x**4)

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Giac [B]  time = 1.1479, size = 331, normalized size = 3.85 \begin{align*} -\frac{1}{12} \,{\left (\frac{12 \, B b^{2} e^{\left (-1\right )}}{x e + d} - \frac{18 \, B b^{2} d e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}} + \frac{12 \, B b^{2} d^{2} e^{\left (-1\right )}}{{\left (x e + d\right )}^{3}} - \frac{3 \, B b^{2} d^{3} e^{\left (-1\right )}}{{\left (x e + d\right )}^{4}} + \frac{12 \, B a b}{{\left (x e + d\right )}^{2}} + \frac{6 \, A b^{2}}{{\left (x e + d\right )}^{2}} - \frac{16 \, B a b d}{{\left (x e + d\right )}^{3}} - \frac{8 \, A b^{2} d}{{\left (x e + d\right )}^{3}} + \frac{6 \, B a b d^{2}}{{\left (x e + d\right )}^{4}} + \frac{3 \, A b^{2} d^{2}}{{\left (x e + d\right )}^{4}} + \frac{4 \, B a^{2} e}{{\left (x e + d\right )}^{3}} + \frac{8 \, A a b e}{{\left (x e + d\right )}^{3}} - \frac{3 \, B a^{2} d e}{{\left (x e + d\right )}^{4}} - \frac{6 \, A a b d e}{{\left (x e + d\right )}^{4}} + \frac{3 \, A a^{2} e^{2}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-3\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)/(e*x+d)^5,x, algorithm="giac")

[Out]

-1/12*(12*B*b^2*e^(-1)/(x*e + d) - 18*B*b^2*d*e^(-1)/(x*e + d)^2 + 12*B*b^2*d^2*e^(-1)/(x*e + d)^3 - 3*B*b^2*d
^3*e^(-1)/(x*e + d)^4 + 12*B*a*b/(x*e + d)^2 + 6*A*b^2/(x*e + d)^2 - 16*B*a*b*d/(x*e + d)^3 - 8*A*b^2*d/(x*e +
d)^3 + 6*B*a*b*d^2/(x*e + d)^4 + 3*A*b^2*d^2/(x*e + d)^4 + 4*B*a^2*e/(x*e + d)^3 + 8*A*a*b*e/(x*e + d)^3 - 3*
B*a^2*d*e/(x*e + d)^4 - 6*A*a*b*d*e/(x*e + d)^4 + 3*A*a^2*e^2/(x*e + d)^4)*e^(-3)